{
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   "cell_type": "code",
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   "metadata": {},
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   "source": [
    "using VMLS\n",
    "using LinearAlgebra\n",
    "using Plots"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Chapter 4\n",
    "# Clustering\n",
    "### 4.1 Clustering\n",
    "### 4.2 A clustering objective"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "In Julia, we can store the list of vectors in a Julia list or tuple of $N$ vectors. If we call this list `x`, we can access the `i`th entry (which is a vector) using `x[i]`. To specify the clusters or group membership, we can use a list of assignments called assignment, where `assignment[i]` is the number of the group that vector `x[i]` is assigned to. (This is an integer between 1 and k.) (In VMLS chapter [4](https://web.stanford.edu/~boyd/vmls/vmls.pdf#chapter.4), we describe the assignments using a vector $c$ or the subsets $Gj$.) We can store the $k$ cluster representatives as a Julia list called `reps`, with `reps[j]` the `j`th cluster representative. (In VMLS we describe the representatives as the vectors `z1, . . . , zk.`)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
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    {
     "data": {
      "text/plain": [
       "Jclust (generic function with 1 method)"
      ]
     },
     "execution_count": 1,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "Jclust(x,reps,assignment) =\n",
    "avg( [norm(x[i]-reps[assignment[i]])^2 for i=1:length(x)] )"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
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    {
     "data": {
      "text/plain": [
       "3-element Array{Array{T,1} where T,1}:\n",
       " Array{Int64,1}[[0, 1], [1, 0], [-1, 1]]\n",
       " Array{Int64,1}[[1, 1], [0, 0]]         \n",
       " [1, 2, 1]                              "
      ]
     },
     "execution_count": 3,
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     "output_type": "execute_result"
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   "source": [
    "x = [ [0,1], [1,0], [-1,1] ]\n",
    "reps = [ [1,1], [0,0] ]\n",
    "assignment = [1,2,1]\n",
    "[x,reps,assignment]"
   ]
  },
  {
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   "execution_count": 10,
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   "outputs": [
    {
     "data": {
      "text/plain": [
       "2.0"
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     "execution_count": 10,
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     "output_type": "execute_result"
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   "source": [
    "Jclust(x,reps,assignment)"
   ]
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   "source": [
    "assignment = [1,1,2];\n",
    "Jclust(x,reps,assignment)"
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 4.3 The k-means algorithm\n",
    "\n",
    "We write a simple Julia implementation of the $k$-means algorithm and apply it to a set of points in a plane, similar to the example in Figure [4.1](https://web.stanford.edu/~boyd/vmls/vmls.pdf#figure.4.1) of VMLS. We first create a function `kmeans` that can be called as `assignment, representatives = kmeans(x, k)` where `x` is an array of $N$ vectors and `k` is the number of groups. The first output argument is an array of `N` integers, containing the computed group assignments (integers from $1$ to $k$). The second output argument is an array of `k` vectors, with the `k` group representatives. We also include two optional keyword arguments, with a limit on the number of iterations and a tolerance used in the stopping condition."
   ]
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   "cell_type": "code",
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   "metadata": {},
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    {
     "data": {
      "text/plain": [
       "kmeans_ (generic function with 1 method)"
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   "source": [
    "function kmeans_(x, k; maxiters = 100, tol = 1e-5)\n",
    "#Overriding the \"error in method definition: function VMLS.kmeans must be explicitly imported to be extended\"\n",
    "#By writing kmeans_ instead of kmeans to display this implementation\n",
    "N = length(x)\n",
    "n = length(x[1])\n",
    "distances = zeros(N) # used to store the distance of each\n",
    "# point to the nearest representative.\n",
    "reps = [zeros(n) for j=1:k] # used to store representatives.\n",
    "    \n",
    " # ’assignment’ is an array of N integers between 1 and k. The initial assignment is chosen randomly.\n",
    "assignment = [ rand(1:k) for i in 1:N ]\n",
    "\n",
    "Jprevious = Inf # used in stopping condition\n",
    "for iter = 1:maxiters\n",
    "\n",
    "    # Cluster j representative is average of points in cluster j.\n",
    "    for j = 1:k\n",
    "        group = [i for i=1:N if assignment[i] == j]\n",
    "                    reps[j] = sum(x[group]) / length(group);\n",
    "                    end;\n",
    "\n",
    "                # For each x[i], find distance to the nearest representative\n",
    "                # and its group index.\n",
    "                for i = 1:N\n",
    "                    (distances[i], assignment[i]) =\n",
    "                    findmin([norm(x[i] - reps[j]) for j = 1:k])\n",
    "                    end;\n",
    "\n",
    "                # Compute clustering objective.\n",
    "                J = norm(distances)^2 / N\n",
    "\n",
    "                # Show progress and terminate if J stopped decreasing.\n",
    "                println(\"Iteration \", iter, \": Jclust = \", J, \".\")\n",
    "                if iter > 1 && abs(J - Jprevious) < tol * J\n",
    "                    return assignment, reps\n",
    "                    end\n",
    "                Jprevious = J\n",
    "                end\n",
    "\n",
    "                end\n"
   ]
  },
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Initialization.** As discussed in VMLS (page [76](https://web.stanford.edu/~boyd/vmls/vmls.pdf#section*.91), the k-means algorithm can start from a random initial choice of representatives, or from a random assignment of the points in `k` groups. In this implementation, we use the second option (`line 11`). The Julia function `rand(1:k)` picks a random number from the set `1:k`, i.e., the integers `1, . . . , k`. On `line 11` we create an array assignment of $N$ elements, with each element chosen by calling `rand(1:k)`. \n",
    "\n",
    "**Updating group representatives.** `Lines 17–20` update the `k` group representatives. In `line 18`, we find the indexes of the points in cluster `j` and collect them in an array group. The expression `x[group]` on `line 19` constructs an array from the subset of elements of `x` indexed by group. The function sum computes the sum of the elements of the array `x[group]`. Dividing by the number of elements length(`x[group]`) gives the average of the vectors in the group. The result is `j`th the group representative. This vector is stored as the `j`th element in an array `reps` of length `N`. \n",
    "\n",
    "**Updating group assignments.** On `lines 24–27` we update the group assignments. The Julia function `findmin` computes both the minimum of a sequence of numbers and the position of the minimum in the sequence. The result is returned as a `2 - tuple`. On `lines 25–26`, we apply `findmin` to the array of `k` distances of point `x[i]` to the k representatives. We store the distance to the nearest representative in `distances[i]`, and the index of the nearest representative (i.e., the new assignment of point `i`) in `assignment[i]`. \n",
    "\n",
    "**Clustering objective.** On `line 30` we compute the clustering objective `Jclust` (equation ([4.1](https://web.stanford.edu/~boyd/vmls/vmls.pdf#equation.4.2.1)) in VMLS) as the square of the RMS value of the vector of distances. \n",
    "\n",
    "**Convergence.** We terminate the algorithm when the improvement in the clustering objective becomes very small (`lines 34–36`). \n",
    "\n",
    "**Example.** We apply the algorithm on a randomly generated set of `N = 300` points, shown in Figure 4.1. These points were generated as follows."
   ]
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    "X = vcat( [ 0.3*randn(2) for i = 1:100 ],\n",
    "[ [1,1] + 0.3*randn(2) for i = 1:100 ],\n",
    "[ [1,-1] + 0.3*randn(2) for i = 1:100 ] )\n",
    "scatter([x[1] for x in X], [x[2] for x in X])\n",
    "plot!(legend = false, grid = false, size = (500,500), xlims = (-1.5,2.5), ylims = (-2,2))"
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   "source": [
    "**Figure 4.1** 300 points in a plane."
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "On the first line we generate three arrays of vectors. Each set consists of 100 vectors chosen randomly around one of the three points `(0, 0)`, `(1, 1)`, and `(1,−1)`. The three arrays are concatenated using `vcat` to get an array of 300 points. \n",
    "\n",
    "Next, we apply our kmeans function and make a figure with the three clusters (Figure 4.2)."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 25,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Iteration 1: Jclust = 0.9696770976109341.\n",
      "Iteration 2: Jclust = 0.3011283508986594.\n",
      "Iteration 3: Jclust = 0.19784093445440018.\n",
      "Iteration 4: Jclust = 0.1794501503595213.\n",
      "Iteration 5: Jclust = 0.1789211624134182.\n",
      "Iteration 6: Jclust = 0.1788830702130446.\n",
      "Iteration 7: Jclust = 0.1788830702130446.\n"
     ]
    },
    {
     "data": {
      "text/plain": [
       "([1, 1, 1, 1, 1, 1, 1, 1, 1, 1  …  3, 3, 3, 3, 3, 3, 3, 3, 1, 3], Array{Float64,1}[[-0.0301011, 0.0509075], [1.02602, 0.99299], [0.983705, -0.977681]])"
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     "execution_count": 25,
     "metadata": {},
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   "source": [
    "k = 3\n",
    "assignment, reps = kmeans(X, k)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 29,
   "metadata": {},
   "outputs": [
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     },
     "execution_count": 29,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "N = length(X)\n",
    "grps = [[X[i] for i=1:N if assignment[i] == j] for j=1:k]\n",
    "scatter([c[1] for c in grps[1]], [c[2] for c in grps[1]])\n",
    "scatter!([c[1] for c in grps[2]], [c[2] for c in grps[2]])\n",
    "scatter!([c[1] for c in grps[3]], [c[2] for c in grps[3]])\n",
    "plot!(legend = false, grid = false, size = (500,500), xlims = (-1.5,2.5), ylims = (-2,2))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "**Figure 4.2** Final clustering\n",
    "\n",
    "### 4.4 Examples\n",
    "\n",
    "### 4.5 Applications"
   ]
  }
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